Proof. Make the antithesis, that there is an integer x0subscriptx0x_{0} such that p⁢(x0)=0psubscriptx00p(x_{0})=0. This x0subscriptx0x_{0} cannot be even, because else all terms of p⁢(x0)psubscriptx0p(x_{0}) except a0subscripta0a_{0} were even and thus the whole sum could not have the even value 0. Consequently, x0subscriptx0x_{0} and also its powers have to be odd. Since